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386 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-29, NO. 4, APRIL 1981

strates a) are accurate to within & 8 percent, b) are simple toapply, c) are not wasteful of material, and d) can determine thepermittivity associated with a specific section of microstrip.

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[2]

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ReferenCeSB, W. .lerws, R. M. Pannell, and J. A. H. Steeden, Attenuation inmicrostrip transmission tines with very 10SSY substrates, in 1980IEEE/ MTT-S Int. Mtcrowaoe Symp. (Washington D.C.), paper V-8,May 1980.H. A. Wheeler, Transmission-line properties of parallel strips separatedby a dielectric sheetj IEEE Trans. M~crowaoe 77reoty Tech., vol. MT1- 13,pp. 172 185, Mar. 1965.W. J. Getsinger, Microstrip dispersion model, IEEE Trans. MtcrowaoeTheory Tech., vol. MTT-21, pp. 34-39, Jan. 1973P. Trough ton, Measurement technique in microstrip, Electron. Lert.,VO]. 5, No. 2, pp. 2526, 1969.I. Wolf and N. Knopplk, Mrcrostrip ring resonator and dmpersionmeasurement on microstrip lines, Electron. Left., vol. 7. no. 26, pp.779-781, 1971.J. G, IUchings, An accurate experimental method for determining theimportant properties of microstrip transmission lines, Marcom Reu.,fourth quarter, pp. 2 I O-216, 1974.B. Easter, The eqmvalent cmcuit of some microstrip dlscontmuitles,IEEE Trans. Mzcrowaue Theory Tech., vol. MTT-23, pp. 655-660, Aug.1975,J. S. Napoli and J. J, Hughes, A simple technique for the accuratedetermination of the microwave dielectric constant for microwave m-integrated circuits, IEEE Trans. Mzcrowave Theo~ Tech , vol. MTT- 19,pp. 664-665, hdy 1971.J. Q. Howell, A qnick, accurate method to measure the dielectricconstant of microwave integrated circuit substrates, IEEE Trans. Mzcro-waoe Theory Tech., vol. MTT-21, pp. 142 143, Mar. 1973.P, H, Ladbrooke, M. H. N, Potok, and E. H. England, Coupling errorsin cawty-resonance measurements on MIC dielectrics, IEEE Trans.Microwave Therny Tech., vol. MfT-21, pp. 560-562, Aug. 1973.E, 0, Hammerstad, Equations for microstrip cmcuit design, in Proc.European Microwave Conf. pp. 268 272, (Hamburg, Germany), Sept.1975

Empirical Relations for Capacitive and InductiveCoupling Coefficients of Coupled Microstrip LinesS. KAL, D. BHATTACHARYA, AND N. B. CHAKRABORTI

,4 bstracf Empirical relations for inductive and capacitive couplingcoefficients are proposed. The functional relationships are based on thephysical mechanism of coupling in microsfiip lines. Values of couplingcoefficients computed from even- and odd-mode impedances and phasevelocities, available in the literature, are compared with those computedfrom the proposed relations. Micros@ip couplers have been designed on thebasis of coupling coefficients to meet the desired coupling and isolation.

L INTRODUCTIONMicrostrip coupled lines are conventionally characterized by

their even- and odd-mode characteristic impedances and phaseconstants [ 1] [3]. These may be computed from a knowledge ofthe even- and odd-mode capacitances [2], [4]. However, there areno simple relations available for finding the parameters involvedin the design. An alternative approach to design may be for-mulated making use of the knowledge of the capacitive andinductive coupling coefficients in coupled lines [5], [6]. Unfor-tunately, here again straightforward formulas are not available.

This paper aims at finding simple empirical relations describ-ing the variation of both capacitive and inductive coupling coeffi-cients with physicaf dimensions of the lines and material parame-

Manuscr]pt received August 26, 1980; revised December 1, 1980.The authors are with the Department of Electronics and Electrical Com-

mumcation Engmeermg, Indian Institute of Technology, Kharagpur 721302,India

mswDIELECTRICSUBS TRATEkr) L\GRouND p~~~~MICROSTRIP COUPLER

(a)

EVEN MODE ODD MOOEELECTRIC FIELD DISTRIBUTION

(b)Fig, 1. (a) Schematic diagram of a microstnp coupler. (b) Electric field

distribution m even- and odd-mode excitation of the microstrip coupledlines

ters of the substrates. The relations permit ready application indesign of coupled systems realized on dielectric or ferromagneticsubstrates. These should be particularly useful in the optimizationprocess in CAD because of the simple mathematical forms.

II. FORMULATION OF THX EMPIRICAL llELATIONSA schematic diagram of a coupled line on a substrate char-

acterized by relative perrnittivit y c, and relative permeability p, isshown in Fig. 1(a). The coupling is basically through fringe fields.The extent of the fringe field determines the electric and mag-net ic coupling coef ficients. The field solution of this type ofstructure outside the region bounded by the top electrode exhibitsan exponentially decaying character [7]. The coupling coefficientsare thus expected to vary exponentially with the character ist icdimensional ratio (5/H). The results given in Milligan [8] ofvariation of mutual and self-capacitances can be used to showthat their ratio varies exponentially with W/H and S/27. It isalso noticed that the variation of capacitive coupling coefficientkc with ~/~ is faster than with W/H. The results of Bryant andWeiss [2] indicate that for low values of dielectric constant thevariation of coupling with c, is rapid while it is rather slow forhigh values of c,. Characteristic impedances and phase velocitiesin even and odd modes may be found from a knowledge of even-and odd-mode capacitances and inductances; the inductance iscomputed from the capacitance of a medium with a relativedielectric constant of unity [4]. This suggests that inductivecoupling coefficient k ~, should be independent of c, and shoulddepend on p,. Incorporating the above ideas the empirical rela-tions for the coupling coefficients kc and k~ may now be writtenas

kc=0.55exp[(AIS/H+ B, W/H)] (1)kl=0.55exp[ -(,42 S/H+ B,W/H)]. (2)

A,, B, are functions of c, and A ~, B2 are functions of p,: .

001 8-9480/8 1/0400-0386$00.75 Q 1981 IEEE

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-29, NO, 4, APRIL 198] 387TABLE IIPERFORMANCEFTESTCOUPLERSTTHECENTERFREOUENCYTABLE ICOMPARISONFTHEVALUESOF kc AND k~ OBTAINED FROMEMPIRICAL RELATIONSWITH THOSECOMPUTEDUSING (4) AND (5) CASE A

Al um$na subsi rate (~, g 61W/H .0836 S/H.0L02

CASE BYIG(Trans Tech G1131subst rat e ( =, ,15 4 )

W/H=0714 S/H=0620p, :03.97at ~=2000Glw

FROM THE RESULTSIN REFERENCES CITEDw $,s,11s,Om lbq ,rm..s[21

[91

[91

[31[11

kc

0386036303170.29602510213

0..49 MO19.X&t Ewtmenl.al W.* Dew Expdm.t.l.0 [- Itw.t,cd .01, .!, . 1 .

COuplmg ,,[n fjB 1098 1030 13 1330

Isobmon 23In dE 2370 2350 20 2350

VSWR - - 1 05 1, 56

I100 02

*lo o of.100 08100 10

02020202

0437 0386 041.80425 0367 04430402 0313 0 Los0391 0.289 0 3S2O 364 0240 03550339 0206 03250 42S 0359 0.4400338 0252 03480.369 0303 04080 211 0103 02110 LLL 0370 0 L26o.f,37 O 361 0 u?O28L 0180 02800338 0236 03340262 0141 02520463 0366 04740363 0233 03760380 0303 0375

100 115 02100 20 02

OL060.911

02040208

034802380286010703880373018602330 lLS

h p,,,,icln S.d .,, ,.w.aN.an, ( 36).(381 .f ,,1,,,.,, [61160160160

010204

020206 3PSZI*16 O 06 O&160 10 06104 0360 0 12C10.L O785 0301

2 5L 228 005

04080272

Lo, ,337S,SS Values of kC and kL for same d,mens!ona[ rot,o but

The numerical coefficients have been chosen such that the result-ing values of kc and k~ agree with those computed from theeven- and odd-mode characteristic impedances (ZO,and ZOO)andphase velocities (BO= and /?OO)quoted by other authors. Values ofkc and k~ can be obtained using relations (22) and (28) ofreference [5] and are given by

z~ ---------------? 10~~u o- 1 I I2 3 L 5 6 7FREQUENCV ( GHz ) THEORETICAL ---- EXPERlt.fENTALkc= Zoeifoo zooPoe

OCBOCJ zOo POe(4)

Fig. 2. Measured performance of a test microstnp coupler on akunma sub-strate (C, =9.6). The solid and dotted lines indicate theoretical and experi-mental values, respect ively.

of the empiricaf relations (1) and (2). The performance of amicrostrip coupler expressed in terms of coupling, directivity, andVSWR are related to kc, kl,, and 00 (electrical length of thecoupled region at center frequency) through the relations given inequations (44) (46) of reference [6].I) Case,4: For a 25-roil thickness alumina substrate (c, = 9.6),the values of kc and k{, needed for a 1l-dB coupler with 23-dBisolation at a center frequency of 5.5 GHz are 0.237 and 0.327,respectively. Using the empirical relations (1) (3), the values forW/H and S/H are found to be 0.836 and 0.402, respectively.The W/H ratio of the uncoupled region corresponds to char-acteristic impedance of 50 Q. It is observed that the value ofcharacteristic impedance of the coupled region calculated byusing the approximate formula Z. = ~- is about 52 ~. Theexperimentally observed VSWR is found to be less than 1.3 overthe frequency range 2 7 GHz. The coupling, isolation, etc., arecompared in Table II with the experimentally obtained data atthe center frequency.2) Case B; For a 25-roil thickness YIG (Trans Tech G11 3)

substrate (t, = 15.4, saturation magnetization= 1780 G), the val-ues of kc and k[, needed for a 13-dB coupler with 20-dB isolationat a center frequency of 7.0 GHz are 0.160 and 0.288, respec-tively. The center frequency has been chosen to be well above ~fi,[11] which is about 5 GHz for YIG (Gl 13) substrate. The relat ivepermeability (p,) is taken to be effective permeability (p.ff ) for

(5)For dielectric substrates p, is unity but for magnetic substrates p,changes with magnetic field and the variation of k~ with appliedmagnetic field can be computed using (2) and (3).

III. NUMERICAL WSULTS AND ILLUSTRATIVE DESIGNTable I presents the values of kc and k. computed using (4)

and (5) along with those obtained using empiricaf relations (1)and (2). An asterisk in the Table I is intended to compare the kcand k ~,values for the same dimensional ratio but different c,. Itmay be noted that kl. remains substantially constant with varia-tion of c, while kc changes with c,. The empirical relations (1)and (2) are found to be in good agreement (within 5 percent) withavailable resul ts for S/H> 0.05 and W/H>O. 1 except for the kcvalues of last three entries. Unfortunately, for small values ofW/H and S/H, results of even- and odd-mode phase velocitiesare not available and therefore no comparison can be made. Thevalue of coupling corresponding to S/H= 0.05 and W/H= 0.1 is0.534 for alumina substrate (c, =9.6). For tighter coupling (S/H

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388 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. rvrTT-29, NO. 4, APRIL 1981

iEislot

!*i~

60 65 70 ?5 80FREQUENCY (GHz) THEORETICAL ---- EXPERIMENTAL

Fig 3. Measured performance of a test microstnp coupler on YIG (TramTech, G113) substrate (c, = 15.4; satrrratlon magnetization= 1780 G; appliedmagnetic field =2000 G). The solid and dotted lines indicate theoretical andexper imen ta l values, respect ively.

magnetization above saturation [1]. As an illustrative case, themagnetic field is chosen to be 2000 G applied transversely to theplane of microstnp lines and for this value of the applied mag-netic field pcff of this substrate at 7 GHz is found to be 0.387 [1],[10]. The values of W/H and S/H for above specifications of thecoupler are found to be 0.714 and 0.620, respect ively. The exper-imentally observed values at the centre frequency are comparedwith the specified values in Table II.

Figs. 2 and 3 give the variations of coupling, isolation, andVSWR of these two couplers over the frequency ranges of investi-gation and indicate close agreement with design values. The smalldeviation may be due to etching errors during fabrication of thedevices using thin film technique. The empincaf relations givenhere for capacitive and inductive coupling coefficients provide a

simple method of designing a microstrip coupler on a dielectricor ferromagnetic substrate.

ACKNOWLEDGMENTThe authors would like to thank Prof. G. S. Sanyal, Head,

Radar and Communication Centre, for extending the measure-ment facilities. The authors also wish to thank Dr. S. K. Lahiriand C. K. Maiti for helpful suggestions in device fabrication.Thanks are also due to N. C. Chakraborti and P. K. Dasgupta forassistance in the expenmentaf work.

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REFERENCESL, Young and H. Sobol, Advances m Microwaves, vol. 8. New York:Academic, 1974, ch. 5, pp. 203-287, ch. 6, pp. 295-318.T. G. Bryant and J. A Weiss, Parameters of microstrrp transmissionlines and coupled paws of microstnp lines: IEEE Trans. MzcrowaoeTheory Tech., vol. MTT-16, pp. 1021-1027, Dec. 1968.L. S. Napoh and J, J. Hughes, Characteristics of coupled microstriplines, RCA Reo., vol. 31, pp. 479-498, Sept. 1970.J. L Smith, The evenand odd mode capacitance parameters for coupledlines in suspendedsubstrate, IEEE Trans. Microwave Theo~ Tech., VOLMTT-19, pp. 424-430, May 1971.J E. Adair and G. I. Haddad, Coupled-mode analysls of non-uniformcoupled transmission lines, IEEE Trans. Mzcrowaoe Theory Tech., vol.MTT-17, pp. 746752, Oct. 1969M, K, Krage and G I Haddad, Characteristics of coupled microstriptransmission lines-I: Coupled mode formulation of inhomogeneous Imes,IEEE Trans. Microwave Theory Tech., vol. MTT- 18, pp. 2 17 222, Apr.I 970.S. B. Cohn, Shielded coupled-strip transmission line, IRE Trans.Mmrowuue Theo~ Tech , vol. MTT-3, pp. 29-38, Ott 1955.T. A. Mdhgan. Dimensions of microstrip coupled lines and interdigltalstructures, IEEE Truns, Mzcrowuve Theoiy Tech., vol. Mfl-25, pp.405-410, May 1977.W. M. Libbey, Theory of active non-reciprocal networks, Ph D. disser-tation, Dept. Electrical Eng. Worcester Polytechmc Inst., Worcester, Ma,1968.R. A. Pucel and D J. Masse, Microstrip propagation on magneticsubstrates Part I: Design theory; Part II: Experiment, IEEE TramM[crowaue Theo~ Tech., vol. MTT-20, pp. 304-308 and pp. 309-313,May 1972.G. R. Harrison et al., Ferromagnetic parts of nricrowave integratedcircuits, IEEE Truns Microwave Theory, Tech., vol MT1- 19, pp. 577-588, July 1971